Methods for controlling glass sheet thickness

ABSTRACT

Methods for controlling thickness variations across the width of a glass ribbon ( 104 ) are provided. The methods employ a set of thermal elements ( 106 ) for locally controlling the temperature of the ribbon ( 104 ). The operating values for the thermal elements ( 106 ) are selected using an iterative procedure in which thickness variations measured during a given iteration are employed in a mathematical procedure which selects the operating values for the next iteration. In practice, the method can bring thickness variations of glass sheets within commercial specifications in just a few iterations, e.g., 2-4 iterations.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a continuation of co-pending U.S. application Ser.No. 12/469,244 filed on May 20, 2009, the contents of which in itsentirety is hereby incorporated by reference.

FIELD

This disclosure relates to the manufacture of glass sheets and, inparticular, to methods for controlling variations in the thickness ofglass ribbons from which glass sheets are produced.

BACKGROUND

U.S. Pat. No. 3,682,609 to Stuart M. Dockerty (the Dockerty patent)describes a system for controlling the thickness distribution across thewidth of a glass ribbon by locally controlling its temperature. To doso, the Dockerty patent uses a pair of refractory plates or walls whoselong axes run parallel to the width of the ribbon. One plate is locatedon each side of the ribbon and the pair are positioned along the lengthof the ribbon above the point where the thickness of the ribbon becomesfixed. The plates are placed relatively close to the ribbon so that theycan absorb heat from the molten glass.

An array of tubes is located behind each plate and oriented so thatfluid (e.g., air) ejected from the tubes impinges on the back of theplate. The fluid flow from each tube is individually controllable. Byadjusting the fluid flow from the tubes, the local temperature on thefront face of the plate can be controlled. This local temperatureaffects the local heat loss, and thus the local temperature, of themolten glass, which, in turn, affects the final thickness distributionacross the width of the ribbon. In practice, the Dockerty system hasproven highly effective in controlling thickness variations across thewidth of glass ribbons and is widely used in the production of glasssheets for such demanding applications as substrates for liquid crystaland organic light emitting diode displays (LCDs and OLEDs).

As currently practiced, the air flow rates in the tubes of the Dockertysystem are adjusted manually by operators. Operators look at a measuredsheet thickness trace and use their experience and judgment, or “feel,”to decide which tubes to adjust, and by how much, to eliminatenon-uniformities in the thickness trace. This reliance on “feel” causesa variety of problems.

For example, when there is a significant change to the process, such asa higher glass flow rate or a different glass composition, substantialtime is often needed during start-up until operators acquire a “feel”for the way the changed process behaves. Furthermore, as thicknessvariation specifications are tightened, there is no way of knowingwhether operator “feel” will be able to meet the new specifications and,if so, how long it will take to do so. Although operator “feel” hasworked in the past, it is unclear if it will be up to the challengesimposed by ever more exacting standards for glass sheets, especiallythose used as substrates for display applications.

More generally, relying on operator “feel” means that new operators mustundergo a learning process before they can make sound judgmentsregarding air flow distributions across the width of the ribbon. Withthe expanding demand for flat screen televisions and monitors, there maycome a time when trained operators becomes a scarce resource limitingthe number of glass making machines that can be in operation at any onetime.

The present disclosure addresses these problems and provides methods forcontrolling the temperature distribution across the width of a glassribbon so that sheet thickness variations are within specificationswithout the need for trained operators who have a “feel” for the system.Rather, it has been found that an iterative process which does not relyon “feel” can be employed to meet thickness specifications using a smallnumber of iterations provided that each iteration is based on amathematical analysis (described below) of the thickness behaviorproduced by the prior iteration.

SUMMARY

A method for producing glass sheets is disclosed which includes:

-   -   (I) producing a glass ribbon (104) having a width;    -   (II) using a plurality of thermal elements (106) to control the        temperature of the ribbon (104) across its width at a location        along the length of the ribbon (104) that is prior to the        location where the thickness of the ribbon (104) becomes fixed,        the thermal elements (106) being distributed across the width of        the ribbon (104) and each element (106) being associated with an        independently adjustable operating variable D_(i); and    -   (III) separating glass sheets from the ribbon (104);

wherein step (II) comprises selecting a set of values for the operatingvariables D_(i) of the thermal elements (106) by:

(a) assigning a sheet thickness response function Δt_(i)(x) to each ofthe thermal elements (106) of the form:

Δt _(i)(x)=func(x,x0_(i) ,w _(i), . . . ),

where x is a location on the sheet, x0_(i) is alocation-of-the-thermal-element parameter, w_(i) is a width-of-effectparameter, and func is a function of at least the variable x and theparameters x0_(i) and w_(i);

(b) selecting values for the x0_(i) and w_(i) parameters;

(c) selecting a set of D_(i) values for the thermal elements (106), theset of D_(i) values being associated with a set of amplitude valuesA_(i);

(d) applying the set of D_(i) values to the thermal elements (106) andproducing at least one glass sheet;

(e) measuring a thickness distribution of at least one glass sheetproduced in step (d);

(f) determining a revised set of A_(i) values by minimizing a functionalof (i) the measured thickness distribution or a derivative thereof, (ii)the set of A_(i) values, and (iii) optionally, a target thicknessdistribution, the functional including a linear superposition of thesheet thickness response functions for the thermal elements (106);

(g) applying a set of D_(i) values associated with the set of A_(i)values determined in step (f) to the thermal elements (106) andproducing at least one glass sheet;

(h) measuring a thickness distribution of the at least one glass sheetproduced in step (g); and

(i) comparing the thickness distribution measured in step (h) with athickness variation criterion and, if necessary, repeating steps (f)through (h), one or more times, until the criterion is satisfied, wherefor each repetition, the measured thickness distribution and the set ofA_(i) values used in the functional which is minimized in step (f) arethose determined in the prior repetition.

The reference numbers used in the above summary are only for theconvenience of the reader and are not intended to and should not beinterpreted as limiting the scope of the invention. More generally, itis to be understood that both the foregoing general description and thefollowing detailed description are merely exemplary of the invention andare intended to provide an overview or framework for understanding thenature and character of the invention.

Additional features and advantages of the invention are set forth in thedetailed description which follows, and in part will be readily apparentto those skilled in the art from that description or recognized bypracticing the invention as described herein. The accompanying drawingsare included to provide a further understanding of the invention, andare incorporated in and constitute a part of this specification. It isto be understood that the various features of the invention disclosed inthis specification and in the drawings can be used in any and allcombinations.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a plot of Gaussian and Gaussian-like functions which can beused as sheet thickness response functions.

FIG. 2 is a plot illustrating the close fit of a Gaussian sheetthickness response function to the thickness changes produced by anindividual thermal element.

FIG. 3 is a plot illustrating the applicability of linear superpositionto the thickness response of two thermal elements.

FIG. 4 is a plot illustrating an exemplary relationship between A valuesand D values.

FIG. 5 is a plot illustrating an exemplary relationship between thew_(i) parameter of a Gaussian sheet thickness response function and Dvalues.

FIG. 6 is a plot illustrating an exemplary comparison after oneiteration between a measured thickness distribution (open data points)and a thickness distribution based on A_(i) values and Gaussian sheetthickness response functions (solid curve). More particularly, the datapoints and curve show thickness changes (Δt values) after one iteration.

FIG. 7 is a schematic diagram of a fusion downdraw system employing aDockerty thickness control system.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

The following discussion is in terms of a fusion downdraw process (alsoknown as a fusion process, an overflow downdraw process, or an overflowprocess), it being understood that the methods disclosed and claimedherein are also applicable to other downdraw processes such as a slotdraw process, as well as to processes that operate horizontally, such asthe float process. In the case of the float process (and some fusionprocesses), the thermal elements will be located on only one side of theglass ribbon, i.e., in the case of a float process, the upper side ofthe ribbon. As fusion apparatus is known in the art, details are omittedso as to not obscure the description of the example embodiments.

As shown in FIG. 7, a typical fusion system 100 employs a formingstructure (isopipe) 101, which includes a cavity 102 which receivesmolten glass. The isopipe includes a root 103 where molten glass fromthe isopipe's two converging sides join together to form ribbon 104.After leaving the root, the ribbon traverses edge rollers 105 and thenone or more sets of pulling rollers (not shown), with the edge rollersbeing used to control the width of the ribbon and the pulling rollsbeing used to apply tension to the ribbon causing it to move downward ata prescribed rate.

Also shown in FIG. 7 is a portion of a Dockerty thickness controlsystem. The system is located adjacent the isopipe's root and affectsthe temperature of the molten glass on the surface of the isopipe aswell as the temperature at the uppermost end of the ribbon. (As usedherein and in the claims, the phrase “a location along the length of theribbon that is prior to the location where the thickness of the ribbonbecomes fixed” includes a location on the forming structure whichproduces the ribbon, whether that location is used alone or incombination with a location downstream of the forming structure.) Asdiscussed above, the Dockerty system includes a refractory plate 108 anda series of tubes 106 which serve to control the local temperature ofthe plate and thus the local temperature of the glass ribbon.

Although the Dockerty plate and tube system produces excellent thicknesscontrol, other thermal control systems can be used in the practice ofthe present disclosure if desired. For example, a series of coolingelements that are individually controllable can be aligned across thewidth of the ribbon. As a further alternative, heating elements can beused either alone or in combination with cooling elements. As usedherein and in the claims, the phrase “thermal elements” is intended toinclude such alternate systems, as well as the original Dockerty systemand variations thereof. The following discussion assumes that adjustableair flows from Dockerty tubes are used as the controllable thermalelements, it being understood that references to this specific type ofthermal element are only for ease of presentation and are not intendedto limit the scope of the disclosure or the claims in any manner. Thecontents of the Dockerty patent (i.e., U.S. Pat. No. 3,682,609) isincorporated herein by reference in its entirety.

As understood by skilled persons, there is a one-to-one correspondencebetween locations across the width of a glass sheet and locations on theglass ribbon which produced the sheet. Also, once the thickness of aglass ribbon becomes fixed, it does not change and thus can bedetermined on sheets after they have been separated from the ribbon.Since measurements on sheets are easier to make than measurements on aribbon, thicknesses are normally determined on a glass sheet, but applyequally to the sheet and the ribbon from which the sheet was separated.Accordingly, as used herein and in the claims, thicknesses, thicknessdistributions, thickness response functions, and the like refer to botha glass sheet and the ribbon from which the sheet was made.

In broad outline, the method of this disclosure uses an iterativeprocess to control the thickness distribution across a glass ribbon (andhence across glass sheets made from the ribbon) wherein individualthermal elements (e.g., air flows from individual Dockerty tubes) areadjusted at each iteration based on a mathematical analysis of thethickness behavior produced by the prior iteration. The iterations arecontinued until the thickness uniformity of glass sheets separated fromthe ribbon is within specifications.

Before performing the iterations, the response of the thickness of theribbon to a single thermal element (e.g., a single Dockerty tube) isdetermined experimentally and then a sheet thickness response functionis assigned to each of the thermal elements so that the mathematicalanalysis employed in the iteration process can be performed. Although anexperimental determination is preferred, the response of the ribbon'sthickness to a single thermal element can also be determined fromprevious experience or a model, e.g., a model based on engineeringcalculations.

If, for example, N thermal elements are used to control the sheetthickness distribution, then for each element, the sheet thicknessresponse Δt_(i)(x) can be written:

Δt _(i)(x)=func(x,x0_(i) ,w _(i), . . . )

where i is the element number (i=1, . . . , N), x is a variableindicating location across the width of the ribbon or, equivalently,across the width of the sheet, x0_(i) is a parameter representing theacross-the-ribbon location of the i^(th) thermal element, w_(i) is aparameter representing the width of the thickness effect of the i^(th)thermal element, and func is a function of the variable x and theparameters x0_(i), w_(i) and possibly other parameters, e.g., the β_(i)parameters used in the Mexican Hat and sinus x over x functionsdiscussed below. It should be noted that Δt_(i)(x) represents the changein thickness from the baseline case where all of the thermal elementsare turned off. In many cases, the parameters x0_(i) and w_(i) will besufficient to describe the thickness response to individual thermalelements, in which case Δt_(t)(x) can be written:

Δt _(i)(x)=func(x,x0_(i) ,w _(i))

Based on experimental studies, it has been found that the sheetthickness response function can be a Gaussian function or aGaussian-like function. In particular, Δt_(i)(x) can be written:

Gaussian:

${\Delta \; {t_{i}(x)}} = {\exp \left( {- \left( \frac{x - {x\; 0_{i}}}{w_{i}} \right)^{2}} \right)}$

Lorentzian:

${\Delta \; {t_{i}(x)}} = \frac{1}{\left( {1 + \left( \frac{x - {x\; 0_{i}}}{w_{i}} \right)^{2}} \right)}$

Modified Lorentzian:

${\Delta \; {t_{i}(x)}} = \frac{1}{\left( {1 + \left( \frac{x - {x\; 0_{i}}}{w_{i}} \right)^{2}} \right)^{\frac{3}{2}}}$

Mexican Hat:

${\Delta \; {t_{i}(x)}} = {\left( {1 + \beta_{i}} \right)\left( {{\exp \left( {- \left( \frac{x - {x\; 0_{i}}}{w_{i}} \right)^{2}} \right)} - {\beta_{i}{\exp \left( {- \left( \frac{x - {x\; 0_{i}}}{w\; 2_{i}} \right)^{2}} \right)}}} \right)}$

Sinus x over x:

${\Delta \; {t_{i}(x)}} = \left\lbrack \begin{matrix}{{1,}\mspace{191mu}} & {{{if}\mspace{14mu} {x}} < 10^{- 6}} \\{\frac{\sin \left( {\frac{2\pi}{w_{i}}x} \right)}{\frac{2\pi}{w_{i}}{x\left( {1 + {\beta_{i}\left( {\frac{2\pi}{w_{i}}x} \right)}^{2}} \right)}},} & {{{if}\mspace{14mu} {x}} \geq 10^{- 6}}\end{matrix} \right.$

FIG. 1 plots these functions for x0_(i)=0 in all cases, w_(i)=2 for allcases except the Mexican Hat where w_(i)=3, and β_(i) equal 1.0 and 0.1for the Mexican Hat and sinus x over x functions, respectively. Thehorizontal axis in this plot and in the plots of FIGS. 2, 3, and 6 is inunits of the spacing between thermal elements. In FIG. 1, the opensquare data points represent the Gaussian function; the open triangles,the modified Lorentzian; the filled circles, the Mexican Hat; and thefilled diamonds, the sinus x over x. The β_(i) parameter has differentmeanings in the Mexican Hat and sinus x over x functions and thus will,in general, have different values when fitted to the same experimentaldata. In particular, in the Mexican Hat, β_(i) controls the depth of thenegative part of the function. In sin(x)/x, increasing β_(i) narrows thefunction's effect and decreases negative-positive oscillations outsidethe main peak of the function. In numerical analysis the sinus x over xfunction is treated differently close to x=0 as indicated above.Analytically, such special handling is not necessary. It should be notedthat in general, the same function is used for all of the thermalelements, although, if desired, different functions could be used fordifferent elements.

FIG. 2 demonstrates the ability of a Gaussian function to accuratelyrepresent the sheet thickness response of a thermal element. In thisfigure, the vertical axis shows the thickness response in microns andthe horizontal axis represents distance in terms of thermal elementnumbers (Dockerty tube numbers). The open data points represent measuredvalues and the solid curve represents a Gaussian fit to the experimentalresults. As illustrated by this figure, the thickness response of aglass ribbon (and thus glass sheets produced from the ribbon) to anindividual thermal element can be accurately modeled by a Gaussianfunction. A similar result can be obtained for the Gaussian-likefunctions, such as those discussed above and illustrated in FIG. 1.

Once a sheet thickness response function has been assigned to each ofthe thermal elements, an experimental determination is made as towhether the resulting thickness change from all the thermal elements canbe represented as a linear superposition of individual thicknessresponses, i.e., whether the overall thickness response Δt(x) can bewritten:

$\begin{matrix}{{\Delta \; {t(x)}} = {\sum\limits_{i = 1}^{N}\; {A_{i}\Delta \; {t_{i}(x)}}}} & (1)\end{matrix}$

where A_(i) is the amplitude of the thickness response corresponding tothe i^(th) thermal element. Note again that Δt(x) is the change from thebaseline case where all of the thermal elements are turned off

FIG. 3 shows the results of such an experimental determination. In thisfigure, the vertical axis shows the thickness response in millimetersand the horizontal axis shows thermal element numbers (Dockerty tubenumbers). For this test, two thermal elements (T1 and T2) were usedwhich were operated at amplitudes of 10 and 20 (arbitrary units). Tobetter illustrate linear superposition, the particular thermal elementschosen for FIG. 3 were 6 elements apart, which corresponded to 6 inchesfor the equipment used. The “x” data points represent modeled results(Gaussian response function) for T1 operated at an amplitude of 10; the“+” data points represent modeled results (Gaussian response function)for T2 operated at an amplitude of 10; the solid line represents themodeled results for the superposition of T1 and T2, each operated at anamplitude of 20; and the open data points represent the experimentallymeasured thickness changes for T1 and T2, each operated at 20. As can beseen in this figure, the experimental data validates the use of linearsuperposition in modeling the thickness response to multiple thermalelements acting simultaneously.

The next relationship employed in the mathematical procedure is thatbetween the value of the amplitude of the thickness response (A_(i)) andthe corresponding thermal element operating variable (D_(i)), i.e., thevalue which is controllable during operation of the glass makingmachine, e.g., the magnitude of air flow through a Dockerty flow tube.FIG. 4 shows experimental data (open data points) illustrating the A_(i)versus D_(i) behavior, where A_(i) is plotted along the vertical axisand D_(i) along the horizontal axis, and both are in arbitrary units.

FIG. 5 further shows that the width parameter w_(i) of a Gaussian sheetthickness response function is substantially independent of the value ofD_(i). In this figure, the vertical axis represents the width parameterobtained by fitting a Gaussian to experimental data (see, for example,FIG. 2) and the horizontal axis represents D values. The open datapoints are experimental measurements and the solid line is an assumedconstant w_(i) value. As can be seen, except for very low D values,unlikely to be frequently used in practice, the w_(i) value issubstantially independent of the strength (D value) of the thermalelement (e.g., Dockerty tube). Similar data to that of FIG. 5 can beobtained for the w_(i) and other parameters used in Gaussian-likefunctions, e.g., those discussed above and illustrated in FIG. 1.

Returning to FIG. 4, as can be seen in this figure, the A values are amonotonous function of the D values. Thus, the data can be fit with afunction ƒ: A_(i)=ƒ(D_(i)) for which the inverse function ƒ⁻¹ exists:D_(i)=ƒ⁻¹(A_(i)). A variety of ƒ functions can be used in the practiceof this disclosure. For example, for the data of FIG. 4, the followingfunction has been found to work successfully;

$A = {{f(D)} = \left\lbrack {\begin{matrix}{{\beta \; D^{\alpha}},} & {{{if}\mspace{14mu} D} < {D\; 0}} \\{{{\gamma \; D} + \delta},} & {{{if}\mspace{14mu} D} \geq {D\; 0}}\end{matrix},} \right.}$

where γ, δ, D0 are independent coefficients and

${\alpha = \frac{\gamma \; D\; 0}{{\gamma \; D\; 0} + \delta}},{\beta = \frac{\gamma \; D\; 0}{\alpha \; D\; 0^{\alpha}}}$

The inverse function is then:

$D = {{f^{- 1}(A)} = \left\lbrack \begin{matrix}{\left( {A/\beta} \right)^{\frac{1}{\alpha}},} & {{{if}\mspace{14mu} A} < {{\gamma \; D\; 0} + \delta}} \\{{\left( {A - \delta} \right)/\gamma},} & {{{if}\mspace{14mu} A} \geq {{\gamma \; D\; 0} + \delta}}\end{matrix} \right.}$

The solid curve in FIG. 4 uses this function with the γ, δ, D0coefficients determined by least squares curve fitting. As can be seen,the function accurately fits the experimental data. As an alternative tousing a function to associate D_(i) values with A_(i) values,interpolation of a data table can also be used for this purpose.

With the foregoing in hand, the sheet thickness can be controlled by thefollowing iterative procedure.

Step 1: Measure a thickness trace t_(meas) ^(k)(x) on a glass sheet (oron a population of glass sheets and then compute an average trace),where k is the iteration number. At start-up, k equals 1.

Step 2: Determine if the measured thickness distribution satisfiesspecifications. If so, no further analysis is required. In such a case,a timer will typically be set so that Step 1 is repeated after aspecified delay. Examples of the types of specifications that can beused include:

$\begin{matrix}{{{\frac{1}{x_{e} - x_{b}}{\int_{x_{b}}^{x_{e}}{\left\lbrack {{t_{meas}^{k}(x)} - {t_{target}(x)}} \right\rbrack^{2}\ {x}}}} < \delta},} & (a)\end{matrix}$

where t_(target) (x) is the desired thickness profile, x_(b), x_(e) arequality area margins, i.e., the locations which include, but aretypically somewhat wider than, the portion of the ribbon that iscommercially acceptable and will ultimately be shipped to customers, andis the acceptable thickness distribution tolerance.

$\begin{matrix}{{{\max\limits_{x \in {\lbrack{x_{b},x_{e}}\rbrack}}\left( {\frac{}{x}\left( {t_{meas}^{k}(x)} \right)} \right)} < \delta},} & (b)\end{matrix}$

where δ is the maximum allowed derivative of the thickness profile.

Step 3: If the measured thickness distribution does not satisfyspecifications, a set of amplitudes {A_(i) ^(k+1)}_(i=1) ^(N), whichwill improve the thickness profile, is determined. The amplitudes areobtained by minimizing a functional based on the measured thicknesses ortheir derivatives.

For example, where measured thicknesses are used, an update to theamplitudes can be obtained by minimizing the following functional:

${\min\limits_{{\{ A_{i}^{k + 1}\}}_{i = 1}^{N}}{\int_{x_{b}}^{x_{e}}{\left( {{t_{meas}^{k}(x)} - {t_{target}(x)} + {\sum\limits_{i = 1}^{N}\; {\left( {A_{i}^{k + 1} - A_{i}^{k}} \right)\Delta \; {t_{i}(x)}}}} \right)^{2}{x}}}},$

where, as above, t_(target)(x) is the desired thickness profile, x_(b),x_(e) are the quality area margins, A_(i) ^(k) are the amplitudescorresponding to the current values of the thermal element variablesA_(i) ^(k)=ƒ(D_(i) ^(k)), and A_(i) ^(k+1) are the new set ofamplitudes. For the first iteration, the A_(i) ^(k) values can, forexample, be all equal to zero if all of the thermal elements are turnedoff (i.e., if all the D_(i) values are zero). Alternatively, some or allof the A_(i) ^(k) values for the first iteration can be non-zero if someor all of the D_(i) values are non-zero based on, for example, pastexperience in controlling sheet thickness variations and/or initialproduction of glass sheets prior to the employment of the methodsdisclosed herein.

Where derivatives of measured thicknesses are used, an update to theamplitudes can be obtained by minimizing the following functional:

$\min\limits_{{\{ A_{i}^{k + 1}\}}_{i = 1}^{N}}\left( {\max\limits_{x \in {\lbrack{x_{b},x_{e}}\rbrack}}\left( {{\frac{}{x}\left( {t_{meas}^{k}(x)} \right)} + {\sum\limits_{i = 1}^{N}\; {\left( {A_{i}^{k + 1} - A_{i}^{k}} \right)\frac{}{x}\left( {\Delta \; {t_{i}(x)}} \right)}}} \right)} \right)$

where the symbols have the same meaning as above.

In either case, the minimization can be performed using standardnumerical techniques, e.g., the minimization can be performed using thestandard SOLVER add-in for MICROSOFT'S EXCEL program.

Step 4: Using ƒ⁻¹, values for the thermal element operating variables(D_(i) ^(k+1)) for the new set of amplitude values (A_(i) ^(k+1)) arecalculated and applied to the thermal element controllers:

D _(i) ^(k+1)=ƒ⁻¹(A _(i) ^(k+1))

In a typical application, a timer will then be set so that Step 1 isrepeated after a specified delay, e.g., after the overall process hasstabilized at the new set of D_(i) values.

In practice, the above procedure has been found to successfully cancelout any measured thickness variations in glass sheets in only a fewiterations, e.g., 2-4 iterations. Although all thickness variations maynot be canceled perfectly, the resulting glass sheets are well withinspecifications. For example, using two iterations, the variation inthickness across a glass ribbon was reduced to 8.6 microns, and with oneadditional iteration, it was reduced to 5.9 microns, well withincommercial specifications. The procedure was tested on different glassmaking machines employing the fusion downdraw process and found to worksuccessfully in all cases and, in particular, was found to worksuccessfully under start-up conditions.

FIG. 6 shows representative data of the ability of the foregoingapproach to model and thus control the thickness variations of glassribbons. The vertical axis in this figure shows Δt(x) values inmillimeters after one iteration and, as indicated above, the horizontalaxis is in terms of thermal element number (Dockerty tube number). Thesolid line shows the model results and the open circles show themeasured thickness values. The fit is quite close, thus allowing theminimization procedure to effectively move the measured data towards thedesired target by selecting a new set of A_(i) values for the nextiteration.

From the foregoing, it can be seen that the advantages of the methods ofthis disclosure include: (a) The procedure for calculating values forthe operating variables of thermal elements brings thickness controlfrom the realm of skilled craft to the realm of rigorous procedure.Therefore, using the procedure makes thickness control consistent fromglass-making-machine to glass-making-machine, and from plant to plant.(b) Using this procedure reduces the amount of time required forbringing thickness variations within limits. (c) The procedure gives arigorous way to assess if a given feature in the thickness trace can becontrolled by using a particular thermal control system (e.g., aDockerty system), and if so, how. (d) The procedure can be used toachieve tighter thickness control than possible using the current “feel”approach. (e) The calculation procedure can be used in model-basedautomatic control of sheet thickness, e.g., as part of overallautomation of the sheet manufacturing process.

A variety of modifications that do not depart from the scope and spiritof the invention will be evident to persons of ordinary skill in the artfrom the foregoing disclosure. For example, although normallyt_(target)(x) will be selected to produce glass sheets having asubstantially uniform thickness, it can also be selected to produceglass sheets whose thickness varies in a prescribed manner across thewidth of the sheet. As just one example, the thickness can increase fromone edge of the sheet to the other. The following claims are intended tocover these and other modifications, variations, and equivalents of thespecific embodiments set forth herein.

1. A method for producing glass sheets comprising: (I) producing a glassribbon having a width; (II) using a plurality of thermal elements tocontrol the temperature of the ribbon across its width at a locationalong the length of the ribbon that is prior to the location where thethickness of the ribbon becomes fixed, said thermal elements beingdistributed across the width of the ribbon and each element having anindependently adjustable operating variable D_(i); and (III) separatingglass sheets from the ribbon; wherein step (II) comprises selecting aset of values for the operating variables D_(i) of the thermal elementsby: (a) assigning a sheet thickness response function Δt_(i)(x) to eachof the thermal elements of the form:Δt _(i)(x)=func(x,x0_(i) ,w _(i), . . . ) where x is a location on thesheet, x0_(i) is a location-of-the-thermal-element parameter, w_(i) is awidth-of-effect parameter, and func is a function of at least thevariable x and the parameters x0₁ and w_(i); (b) selecting values forthe x0₁ and w_(i), parameters; (c) selecting a set of D_(i) values forthe thermal elements, said set of D_(i) values being mathematicallyassociated with a set of amplitude values A_(i); (d) applying the set ofD_(i) values to the thermal elements and producing at least one glasssheet; (e) measuring a thickness distribution of at least one glasssheet produced in step (d); (f) determining a revised set of A_(i)values by minimizing a functional of (i) the measured thicknessdistribution or a derivative thereof, (ii) the set of A_(i) values, and(iii) a target thickness distribution, said functional including alinear superposition of the sheet thickness response functions for thethermal elements or derivatives thereof, and said minimization beingperformed numerically using a computer program; (g) applying a set ofD_(i) values mathematically associated with the set of values determinedin step (f) to the thermal elements and producing at least one glasssheet; (h) measuring a thickness distribution of the at least one glasssheet produced in step (g); and (i) comparing the thickness distributionmeasured in step (h) with a thickness variation criterion and, if thecriterion is not satisfied, repeating steps (f) through (h), one or moretimes, until the criterion is satisfied, where for each repetition, themeasured thickness distribution and the set of A_(i) values used in thefunctional which is minimized in step (f) are those determined in theprior repetition; wherein the thickness variation criterion used in step(i) is selected from the group consisting of: (i) a uniform thicknesswithin a predetermined tolerance defined by a commercial specification;and (ii) a predetermined non-uniform thickness within a predeterminedtolerance defined by a commercial specification.
 2. The method of claim1 wherein in step (i), the comparison of the thickness distributionmeasured in step (h) with the thickness variation criterion comprisesdetermining a least squares difference between the measured thicknessdistribution and a target thickness distribution.
 3. The method of claim1 wherein in step (i), the comparison of the thickness distributionmeasured in step (h) with the thickness variation criterion comprisesdetermining a maximum slope of the measured thickness distribution. 4.The method of claim 1 wherein the sheet thickness response function is aGaussian function.
 5. The method of claim 1 wherein the sheet thicknessresponse function is a Gaussian-like function.
 6. The method of claim 5wherein the Gaussian-like function is selected from the group consistingof Lorentzian, modified Lorentzian, Mexican hat, and sinus x over xfunctions.
 7. The method of claim 1 wherein the D_(i) values aremathematically associated with the A_(i) values by a function of theform: $A = {{f(D)} = \left\lbrack {\begin{matrix}{{\beta \; D^{\alpha}},} & {{{if}\mspace{14mu} D} < {D\; 0}} \\{{{\gamma \; D} + \delta},} & {{{if}\mspace{14mu} D} \geq {D\; 0}}\end{matrix},} \right.}$ where γ, δ, D0 are independent coefficients and${\alpha = \frac{\gamma \; D\; 0}{{\gamma \; D\; 0} + \delta}},{\beta = {\frac{\gamma \; D\; 0}{\alpha \; D\; 0^{\alpha}}.}}$8. The method of claim 1 wherein the number of repetitions of steps (f)through (h) required to satisfy the thickness variation criterion ofstep (i) is between two and four.